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ON THE SUPERCRITICALLY DIFFUSIVE MAGNETO-GEOSTROPHIC EQU ATIONS

SUSAN FRIEDLANDER, WALTER RUSIN, AND VLAD VICOL

ABSTRACT. We address the well-posedness theory for the magento-geostrophic equation, namely an activescalar equation in which the divergence-free drift velocity is one derivative more singular than the active scalar.In the presence of supercritical fractional diffusion given by (−∆)γ with 0 < γ < 1, we discover that forγ > 1/2 the equations are locally well-posed, while forγ < 1/2 they are ill-posed, in the sense that there is noLipschitz solution map. The main reason for the striking loss of regularity whenγ goes below1/2 is that theconstitutive law used to obtain the velocity from the activescalar is given by an unbounded Fourier multiplierwhich is both even and anisotropic. Lastly, we note that the anisotropy of the constitutive law for the velocitymay be explored in order to obtain an improvement in the regularity of the solutions when the initial data andthe force have thin Fourier support, i.e. they are supportedon a plane in frequency space. In particular, for suchwell-prepared data one may prove the local existence and uniqueness of solutions for all values ofγ ∈ (0, 1).

CONTENTS

1. Introduction 12. Preliminaries 43. The regime1/2 < γ < 1: well-posedness results 54. The regime0 < γ < 1/2: ill-posedness results 75. The regimeγ = 1/2: a dichotomy in terms of the size of the data 136. Improvement in regularity for “well-prepared” data and source 15Appendix A. Smooth solutions to a linear equation for data with “thin” Fourier support 20References 23

1. INTRODUCTION

The geodynamo is the process by which the Earth’s magnetic field is created and sustained by the mo-tion of the fluid core which is composed of a rapidly rotating,density stratified, electrically conductingfluid. Recently Moffat [19] proposed a model for the geodynamo which is a reduction of the full magne-tohydrodynamic system of PDE. This model can be viewed as a nonlinear active scalar equation in threedimensions. An explicit operatorM [Θ] encodes the physics of the geodynamo and produces the divergencefree drift velocityU(t,x) from the scalar “buoyancy” fieldΘ(t,x). We call this active scalar equation themagnetogeostrophic equation(MG). It has some features in common with the much studied two dimen-sional surface quasigeostrophic equation(SQG) (see [2, 3, 4, 5, 6, 14, 16, 21, 26] and references therein).However the(MG) equation has a number of novel and distinctive features due to the strongly singular andanisotropic nature of the operatorM .

Thecritically diffusive(MG) equation is defined by the following system

∂tΘ+U · ∇Θ = S + κ∆Θ (1.1)

∇ ·U = 0 (1.2)

Date: October 7, 2011.2010Mathematics Subject Classification.76D03, 35Q35, 76W05.Key words and phrases.Magneto-geostrophic models, supercritical diffusion, active scalar equations, Hadamard ill-posedness,

thin Fourier support, anisotropic constitutive law, continued fractions.1

2 SUSAN FRIEDLANDER, WALTER RUSIN, AND VLAD VICOL

whereκ ≥ 0 is a physical parameter, andS(t,x) is aC∞-smooth, bounded in time source term. ThevelocityU is divergence-free, and is obtained fromΘ via

Uj =MjΘ, (1.3)

for all j ∈ 1, 2, 3, where theMj are Fourier multiplier operators with symbols given explicitly by

M1(k) =2Ωk2k3|k|2 − (β2/η)k1k

22k3

4Ω2k23 |k|2 + (β2/η)2k42(1.4)

M2(k) =−2Ωk1k3|k|2 − (β2/η)k32k34Ω2k23 |k|2 + (β2/η)2k42

(1.5)

M3(k) =(β2/η)k22(k

21 + k22)

4Ω2k23|k|2 + (β2/η)2k42(1.6)

where the Fourier variablek ∈ Zd is such thatk3 6= 0. Note thatMj(k) is not defined onk3 = 0, sincefor the self-consistency of the model it is assumed thatΘ andU have zerovertical mean. It can be directlychecked thatkj · Mj(k) = 0 for all k ∈ Zd \ k3 = 0, and hence the velocity fieldU given by (1.3) isindeed divergence-free. The physical parameters of the geodynamo are the following:Ω is the rotation rateof the Earth,η is the magnetic diffusivity of the fluid core,κ is the thermal diffusivity, andβ is the strengthof the steady, uniform mean part of the magnetic field in the fluid core. The perturbation magnetic fieldvectorb(t, x) is computed fromΘ(t,x) via the operator

bj = (β/η)(−∆)−1∂2MjΘ, for all j ∈ 1, 2, 3. (1.7)

We observe that the presence of the underlying magnetic field, reflected in the parameterβ2/η, produces anon-isotropic structure in the symbolsMj .

It is important to note that although the symbolsMi are0-order homogenous under the isotropic scalingk → λk, due to their anisotropy the symbolsMi are not bounded functions ofk. To see this, note thatwhereas in the region of Fourier space wherek1 ≤ maxk2, k3 the Mi are bounded by a constant, uni-formly in |k|, this is not the case on the “curved” frequency regions wherek3 = O(1) andk2 = O(|k1|r),with 0 < r ≤ 1/2. In such regions the symbols are unbounded, since as|k1| → ∞ we have

|M1(k1, |k1|r, 1)| ≈ |k1|r, |M2(k1, |k1|r, 1)| ≈ |k1|, |M3(k1, |k1|r, 1)| ≈ |k1|2r. (1.8)

In fact, it may be shown that|M(k)| ≤ C|k|, whereC(β, η,Ω) > 0 is a fixed constant, and this boundis sharp. The fact that symbols are at most linearlyunboundedin the whole of Fourier space implies thatM [Θ] is thederivativeof a singular integral operator acting onΘ (see [9] for details), as opposed to thecase of the(SQG) equation whereU is the Riesz transform ofΘ.

In [9] we studied a class of nonlinear active scalar equations of which the(MG) system (1.1)–(1.6) is amember, namely (1.1) with

Uj = ∂jTijΘ (1.9)

for all j ∈ 1, 2, 3 whereTij is a3 × 3 matrix of Calderon-Zygmund operators of convolution typesuchthat ∂i∂jTijφ = 0 for any smooth functionφ. Inspired by the proof of Caffarelli and Vasseur [2] for theglobal well-posedness of the critically diffusive(SQG) equations, we used DeGiorgi techniques to obtainthe global well-posedness of the critically diffusive(MG) equations, namely (1.1)–(1.6). We remark thatfor both the diffusive(MG) and(SQG) equationscriticality is defined with respect to theL∞ maximumprinciple. In [10] we considered the non-diffusive versionof the(MG) equations, namely (1.1)–(1.6) withκ = 0. In contrast with the critically diffusive problem where the (MG) equation is globally well-posedand the solutions areC∞ smooth for positive time, we proved that whenκ = 0 the equation is Hadamardill-posed in any Sobolev space1 which embeds inW 1,4.

1The non-diffusive(MG) equations are however locally well-posed in spaces of real-analytic functions, cf. [10].

SUPERCRITICALLY DIFFUSIVE(MG) 3

In this present paper we study the fractionally diffusive version of the(MG) equation, namely

∂tΘ+U · ∇Θ = S − κ(−∆)γΘ (1.10)

∇ ·U = 0, U = M [Θ] (1.11)

with Uj = Mj[Θ] where the symbolsMj are given by (1.4)–(1.6) and the parameterγ ∈ (0, 1). The mainobservation is that the structure ofMj (anisotropy, unboundedness, and evenness) and theweaksmoothingeffects of the nonlocal operator(−∆)γ combine to produce a sharp dichotomy across the valueγ = 1/2.More precisely, ifγ > 1/2 the equations are locally well-posed, while ifγ < 1/2 they are ill-posed inSobolev spaces, in the sense of Hadamard.

In section 3 we use energy estimates to prove that forγ ∈ (1/2, 1) the fractionally diffusive(MG)equations are locally well-posed in Sobolev spacesHs(T3) for s > 5/2+(1−2γ). With an additional smalldata assumption we obtain a global well-posedness result. To see why one may not use energy estimatesto obtain local well-posedness of the(MG) equations whenγ < 1/2, we point out that in theHs energyestimate for (1.10) there are only two terms for which more than “s derivatives” fall on a single function:

Tbad =

∫Λs

U · ∇Θ ΛsΘ and Tgood =

∫U · ∇ΛsΘ ΛsΘ (1.12)

where we denotedΛ = (−∆)1/2. Since∇ · U = 0, upon integrating by parts we haveTgood = 0. Onthe other hand, the termTbad does not vanish in general. The diffusion competing withTbad is given byκ‖Λs+γΘ‖2L2 . Two issues arise. First, sinceU ≈ ΛΘ andγ < 1/2, and one cannot control‖Λs−γ

U‖L2

with ‖Λs+γΘ‖L2 , and therefore boundingTbad could only be achieved by exploring some extra cancellation.Second, since the symbolsMj are even the operatorM is notanti-symmetric, thus one cannot re-writeTbadas−

∫ΛsΘ·M [∇Θ ΛsΘ] (note that if one could do this, a suitable commutator estimate of Coifman-Meyer

type would close the estimates at the level of Sobolev spaces, as in [3, 10]). Hence there is no a commutatorstructure inTbad, and the estimates do not seem to close at the level of Sobolevspaces when0 < γ < 1/2.

In section 4 we consider the caseγ ∈ (0, 1/2) and prove that the solution map associated with theCauchy problem isnot Lipschitz continuous with respect to perturbations in the initial data in the topologyof a certain Sobolev spaceX. Hence the Cauchy problem is ill-posed in the sense of Hadamard. This isachieved by considering a specific steady profileΘ0, and constructing functions which are “close” toΘ0, butwhich in arbitrarily short time deviate arbitrarily far from Θ0, when measured inX. The arguments hingeon using techniques of continued fractions to exhibit an unstable eigenvalue for the linearized equation(cf. [8, 18]). These techniques produce alower bound on the growth rate of eigenvalues to the linearizedequations, and in the case whereγ ∈ (0, 1/2) we prove that the magnitude of this lower bound can be madearbitrarily large. Once unstable eigenvalues of arbitrarily large part have been obtained for the linearizedequations, the Lipschitz ill-posednedness of the full non-linear equations may be obtained using classicalarguments (see, e.g. [10, 11, 12, 20, 24]). We emphasize thatthe mechanism producing ill-posedness is notmerely the order one derivative loss in the mapΘ → U . Rather, it is the combination of the derivative losswith the anisotropy of the symbolM and the fact that this symbol iseven. We note that the even natureof the symbol ofM plays a central role in the proof of the non-uniqueness givenin [22] for theL∞-weaksolutions of the non-diffusive(MG) equations.

In section 5 we study the more subtle transitional case whenγ = 1/2. In this case, if the initial data is“small” with respect toκ, we use energy estimates to prove that there exists a unique global solution. Indramatic contrast, there exists initial data which is “large” with respect toκ, and for which the associatedlinear operator has arbitrarily large unstable eigenvalues, which may be use to prove that the equations areHadamard ill-posed. Situations where the qualitative behavior of the solution depends crucially on the normof initial data are also encountered for other evolution equations. For example, in the two-dimensionalKeller-Segel model of chemotaxis, properties of the solution are strongly dependent on the total massmof cells. If m < 8π, a global bounded solution of the initial value problem exists and is dispersed fort → ∞, while form > 8π, blow-up in finite time occurs. The critical casem = 8π is by now understood

4 SUSAN FRIEDLANDER, WALTER RUSIN, AND VLAD VICOL

and the result states that a global solution exists and possibly becomes unbounded ast → ∞ (see [13]and references therein). We also mention that for the Rayleigh-Taylor and the Helmholtz problems it is“geometric” conditions, rather than merely the size of the data, which leads to ill-posedness (see, for instance[7]).

In section 6 we take advantage of the anisotropy of the symbols (1.4)–(1.6) and observe an interestingphenomenon: when the initial data and the source term are “well-prepared” in an appropriate sense, it ispossible to prove stronger regularity results for the ensuing solutions. More specifically, when the initialdata and the force have “thin” Fourier support, i.e. they aresupported purely on a planePq = k =(k1, k2, k3) ∈ Z3 : k2 = qk1 in frequency space, whereq is a fixed non-zero rational number, the operatorM behaves like an operator of order zero. The reason is that theFourier symbolsMj are unbounded onlyon curved regions in frequency space (cf. (1.8) above), but if k ∈ Pq theMj(k) are bounded by a constantdepending onq. This observation, combined with the fact that when the dataand source have frequencysupport onPq, then so does the solution of the(MG) equations at all later times, means that the smoothingproperties of the fractional diffusion term are stronger than they are for the generic data situation. Henceit is possible to prove stronger regularity results, and in particular the local existence and uniqueness ofSobolev solutions holdsfor all values ofγ ∈ (0, 1). We note that this is not in contradiction with our resultsproven in sections 4 and 5, since in order to obtain ill-posedness we need to sendq → 0 (so that we obtaineigenvalues of the linear operator with arbitrarily large real part), whereas in the case of well-prepared datathe value ofq > 0 is fixed. Other uses of thin Fourier support for different problems can be found, e.g. in[1, 24].

2. PRELIMINARIES

This section contains a few auxiliary results used in the paper. In particular, we recall the, by nowclassical, product and commutator estimates, as well as theSobolev embedding inequalities. Proofs of theseresults can be found for instance in [15, 23, 24]. Let us denoteΛ = (−∆)1/2.

Proposition 2.1(Product estimate). If s > 0, then for allf, g ∈ Hs ∩ L∞, and we have the estimates

‖Λs(fg)‖Lp ≤ C (‖f‖Lp1‖Λsg‖Lp2 + ‖Λsf‖Lp3‖g‖Lp4 ) , (2.1)

where1/p = 1/p1 + 1/p2 = 1/p3 + 1/p4, andp, p2, p3 ∈ (1,∞). In particular

‖Λs(fg)‖L2 ≤ C (‖f‖L∞‖Λsg‖L2 + ‖Λsf‖L2‖g‖L∞) . (2.2)

In the case of a commutator we have the following estimate.

Proposition 2.2(Commutator estimate). Suppose thats > 0 andp ∈ (1,∞). If, f, g ∈ S, then

‖Λs(fg)− fΛsg‖Lp ≤ C(‖∇f‖Lp1‖Λs−1g‖Lp2 + ‖Λsf‖Lp3‖g‖Lp4

)(2.3)

wheres > 0, 1/p = 1/p1 + 1/p2 = 1/p3 + 1/p4, andp, p2, p3 ∈ (1,∞).

We shall use as well the following Sobolev inequality.

Proposition 2.3(Sobolev inequality). Suppose thatq > 1, p ∈ [q,∞) and1/p = 1/q− s/d. Suppose thatΛsf ∈ Lq, thenf ∈ Lp and there is a constantC > 0 such that

‖f‖Lp ≤ C‖Λsf‖Lq . (2.4)

Throughout this paper we shall make use the following definition of solutions of (1.1)–(1.3).

Definition 2.4 (Solution of the (MG) equation). Let s ≥ 1/2, T > 0, κ > 0, andγ ∈ (0, 1). GivenΘ0 ∈ Hs(T3), we call a function

Θ ∈ L∞(0, T ;Hs(T3)) ∩ L2(0, T ;Hs+γ(T3)) (2.5)

SUPERCRITICALLY DIFFUSIVE(MG) 5

a solutionof the Cauchy problem(1.1)–(1.3), if∫

T3

(Θ(t, ·)−Θ0)ϕdx+ κ

∫ t

0

∫

T3

Θ(−∆)γϕdxds−∫ t

0

∫

T3

Θ U · ∇ϕdxds = 0 (2.6)

holds for all test functionsϕ ∈ C∞0 (T3), and all t ∈ (0, T ).

3. THE REGIME 1/2 < γ < 1: WELL-POSEDNESS RESULTS

It has been shown in [9] that the system (1.1)–(1.6) is globally well-posed whenγ = 1. In this sectionwe prove that in the rangeγ ∈ (1/2, 1) equations (1.1)–(1.6) are locally well-posed inHs(T3). With anadditional assumption on the size of the initial data and thesource term, we obtain a global well-posednessresult.2

Regarding arbitrarily large initial data, we obtain the following result.

Theorem 3.1(Local existence).Letγ ∈ (1/2, 1), and fixs > 5/2 + (1− 2γ). Assume thatΘ0 ∈ Hs(T3)andS ∈ L∞(0,∞;Hs−γ(T3)) have zero-mean onT3. Then there exists a timeT > 0 and a unique smoothsolution

Θ ∈ L∞(0, T ;Hs(T3)) ∩ L2(0, T ;Hs+γ(T3)) (3.1)

of the Cauchy problem(1.10)–(1.11).

Proof. We multiply equation (1.1) byΛ2sΘ, integrate by parts to obtain the followingHs-energy inequality

1

2

d

dt‖ΛsΘ‖2L2 + κ‖Λs+γΘ‖2L2 ≤

∣∣∣∣∫

R3

SΛ2sΘ

∣∣∣∣+∣∣∣∣∫

R3

U · ∇ΘΛ2sΘ

∣∣∣∣ . (3.2)

We estimate the first term on the right side by∣∣∣∣∫

R3

SΛ2sΘ

∣∣∣∣ ≤ ‖Λs−γS‖L2‖Λs+γΘ‖L2 ≤ 1

2κ‖Λs−γS‖2L2 +

κ

2‖Λs+γΘ‖2L2 . (3.3)

To handle the second term we proceed as follows. First note that∣∣∣∣∫

R3

U · ∇ΘΛ2sΘ

∣∣∣∣ =∣∣∣∣∫

R3

Λs−γ(U · ∇Θ)Λs+γΘ

∣∣∣∣ ≤ ‖Λs−γ(U · ∇Θ)‖L2‖Λs+γΘ‖L2 . (3.4)

The estimate of the product term follows from Proposition 2.1. Hence, we have

‖Λs−γ(U · ∇Θ)‖L2 ≤ C(‖Λs−γU‖Lp‖∇Θ‖

L2pp−2

+ ‖Λs−γ∇Θ‖Lp‖U‖L

2pp−2

) (3.5)

for some2 < p < ∞ to be chosen later. Recall thatUj = MjΘ, where the symbol of the multiplierMj

enjoys a uniform bound|Mj(k)| ≤ C|k|, for j = 1, 2, 3. ThereforeΛ−1Mj are bounded operators onLp

(see [9, Section 4] for details) and we obtain

‖Λs−γU‖Lp ≤ C‖Λs−γ+1Θ‖Lp (3.6)

and

‖U‖L

2pp−2

≤ C‖ΛΘ‖L

2pp−2

. (3.7)

Hence, the product term is bounded by

‖Λs−γ(U · ∇Θ)‖L2 ≤ C‖ΛΘ‖L

2pp−2

‖Λs−γ+1Θ‖Lp . (3.8)

We now fix an arbitraryp such that

3

s− 1< p <

6

2(1− 2γ) + 3=

6

5− 4γ.

2For our convenience, we choose to work in sub-critical Sobolev spaces. The following proofs can be also carried out in thesetting of critical Besov spacesBs

2,1 or in generalBsp,q .

6 SUSAN FRIEDLANDER, WALTER RUSIN, AND VLAD VICOL

Note thatp > 2 sinces > 3/2, and the range forp is non-empty sinces > 5/2+(1−2γ). Forγ ∈ (1/2, 1),our choice ofp and the Sobolev embedding (Proposition 2.3) gives

‖Λs−γ+1Θ‖Lp ≤ ‖ΛsΘ‖1−ζL2 ‖Λs+γΘ‖ζ

L2 (3.9)

whereζ ∈ (0, 1) may be computed explicitly fromγζ = 3/2 − 3/p + 1− γ. In addition, sinceΘ has zeromean, andp > 3/(s − 1), from the Sobolev embedding we obtain

‖ΛΘ‖L

2pp−2

≤ C‖ΛsΘ‖L2 . (3.10)

Combining estimates (3.2)-(3.10) gives

d

dt‖ΛsΘ‖2L2 + κ‖Λs+γΘ‖2L2 ≤ 1

κ‖Λs−γS‖2L2 + C‖ΛsΘ‖2−ζ

L2 ‖Λs+γΘ‖1+ζL2 (3.11)

where0 < ζ < 1 is as defined earlier. The second term on the right side of (3.11) is bounded using theǫ-Young inequality as

κ

2‖Λs+γΘ‖2L2 + Cκ−

2(1+ζ)1−ζ ‖ΛsΘ‖

2(2−ζ)1−ζ

L2 . (3.12)

and we finally obtain the following estimate

d

dt‖ΛsΘ‖2L2 +

κ

2‖Λs+γΘ‖2L2 ≤ 1

κ‖Λs−γS‖2L2 + Cκ

− 2(1+ζ)1−ζ ‖ΛsΘ‖

2(2−ζ)1−ζ

L2 . (3.13)

Using Gronwall’s inequality, from estimate (3.13) we may deduce the existence of a positive timeT =T (‖S‖L∞(0,T ;Hs−γ), ‖Θ0‖Hs , κ) such thatθ ∈ L∞(0, T ;Hs(T3)) ∩ L2(0, T ;Hs+γ(T3)). Note that wehave2(2 − ζ)/(1− ζ) > 2, and hence we may not obtain the global existence of solutions from the energyestimate (3.13), if the initial data has largeHs norm. These a priori estimates can be made formal using astandard approximation procedure. We omit further details.

The second main result of this section concerns global well-posedness in case of small initial data.

Theorem 3.2 (Global existence for small data).Let γ and S be as in the statement of Theorem 3.1,and letΘ0 ∈ Hs(T3) have zero-mean onT3, wheres > 5/2 + (1 − 2γ). There exists a small enoughconstantε > 0 depending onκ, such that if‖Θ0‖αL2‖Θ0‖1−α

Hs + ‖Θ0‖αL2‖S‖1−αL∞(0,∞;Hs−γ)

≤ ε, where

α = 1− (5/2+1−2γ)/s, then the unique smooth solutionΘ of the Cauchy problem(1.10)–(1.11)is globalin time, i.e.

Θ ∈ L∞(0,∞;Hs(T3)). (3.14)

Proof. We proceed as in the proof of Theorem 3.1. The product term in (3.5) is now estimated by

‖Λs−γ(U · ∇Θ)‖L2 ≤ C(‖Λs−γU‖Lp‖∇Θ‖

L2pp−2

+ ‖Λs−γ∇Θ‖Lp‖U‖L

2pp−2

), (3.15)

where

p =6

2(1 − 2γ) + 3

so that

‖Λs+1−γΘ‖Lp ≤ C‖Λs+γΘ‖L2 . (3.16)

With this choice ofp and the above embedding, the product estimate gives us

‖Λs−γ(U · ∇Θ)‖L2 ≤ C‖∇Θ‖L

2pp−2

‖Λs+γΘ‖L2 . (3.17)

Combining (3.17) with (3.2) and proceeding as in (3.13) we obtain

1

2

d

dt‖ΛsΘ‖2L2 + κ‖Λs+γΘ‖2L2 ≤ ‖Λs−γS‖L2‖Λs+γΘ‖L2 + C‖∇Θ‖

L2pp−2

‖Λs+γΘ‖2L2 , (3.18)

SUPERCRITICALLY DIFFUSIVE(MG) 7

which in turn implies

1

2

d

dt‖ΛsΘ‖2L2 +

κ

2‖Λs+γΘ‖2L2 ≤ 1

2κ‖Λs−γS‖2L2 + C‖∇Θ‖

L2pp−2

‖Λs+γΘ‖2L2 . (3.19)

Observe that

‖∇Θ‖L2p/(p−2) ≤ C‖Θ‖αL2‖ΛsΘ‖1−αL2 , (3.20)

whereα = 1− (5/2 + 1− 2γ)/s. Therefore, if

‖Θ‖αL2‖ΛsΘ‖1−αL2 ≤ κ

4C, (3.21)

estimate (3.19), combined with the Poincare inequality‖Λs+γΘ‖L2 ≥ ‖ΛsΘ‖L2 , shows that

d

dt‖ΛsΘ‖2L2 +

κ

2‖ΛsΘ‖2L2 ≤ 1

κ‖S‖2

L∞

t Hs−γx

. (3.22)

and hence

‖ΛsΘ(t, ·)‖2L2 ≤ ‖ΛsΘ0‖2L2 +2

κ2‖S‖2

L∞

t Hs−γx

(3.23)

for all t > 0. Note that taking theL2-product of the equation withΘ gives for anyt > 0

‖Θ(t, ·)‖2L2 + κ

∫ t

0‖ΛγΘ‖2L2 dτ ≤ ‖Θ0‖2L2 , (3.24)

which gives us a basic uniform estimate ofΘ in L∞t L

2x. Hence, from (3.23) and (3.24) we obtain that

condition (3.21) is satisfied for allt > 0 as long as we have

‖Θ0‖αL2‖ΛsΘ0‖1−αL2 + ‖Θ0‖αL2‖Λs−γS(τ, ·)‖1−α

L∞(0,∞;L2)< ǫ, (3.25)

whereǫ is sufficiently small, thereby concluding the proof of the theorem.

4. THE REGIME 0 < γ < 1/2: ILL -POSEDNESS RESULTS

We now turn to the situation whenγ ∈ (0, 1/2), κ > 0, and prove that there exists an example of aninitial datum and time independent force for which it is possible to prove that the fractionally diffusive(MG) equation is ill-posed in the sense of Hadamard. The arguments by which we prove this follow thelines of the proof in the case whenκ = 0 in [10]. We sketch here the proof ofγ ∈ (0, 1/2), but as we shallprove below in Section 5.1, the result is also true whenγ = 1/2 and the initial data islarge in a suitablesense.

4.1. Linear ill-posedness inL2. The classical approach to Hadamard ill-posedness for non-linear problemsis to first linearize the equations about a suitable steady stateΘ0, and then prove that the resulting linearoperatorL has eigenvalues with arbitrarily large real part in the unstable region (see for instance [10, 11,12, 20], and references therein). LetΘ0 = a sin(mx3) for some positive amplitudea and integer frequencym ≥ 1, and defineS = κam2γ sin(mx3). It is not hard to verify thatΘ0 is indeed a steady state of(1.1)–(1.2). We consider thelinear evolution of the perturbationθ = Θ − Θ0, obtained from linearizing(1.1)–(1.2)

∂tθ + am cos(mx3)M3θ + κ(−∆)γθ = 0, (4.1)

where we recall thatM3 is the Fourier multiplier with symbol defined in (1.6). The following theorem statesthat the linear equation (4.1) is Hadamard Lipschitz ill-posed inL2.

Theorem 4.1(Linear ill-posedness).The Cauchy problem associated to the linear evolution

∂tθ = Lθ (4.2)

8 SUSAN FRIEDLANDER, WALTER RUSIN, AND VLAD VICOL

where the linear operatorL and the steady stateΘ0 are given by

Lθ(x, t) = −M3θ(x, t) ∂3Θ0(x3)− κ(−∆)γθ (4.3)

Θ0(x3) = a sin(mx3) (4.4)

for somea > 0, integerm ≥ 1, andγ ∈ (0, 1/2], is ill-posed in the sense of Hadamard overL2. Moreprecisely, for anyT > 0 and anyK > 0, there exists a real-analytic initial dataθ(0,x) such that theCauchy problem associated to(4.2)–(4.4)has no solutionθ ∈ L∞(0, T ;L2) satisfying

supt∈[0,T )

‖θ(t, ·)‖L2 ≤ K‖θ(0, ·)‖Y (4.5)

whereY is any Sobolev space embedded inL2.

As we mentioned above, in order to prove Theorem 4.1 we construct a sequence of eigenvaluesσ(j) ofthe operatorL, which diverge to∞ asj → ∞. For this purpose, given anyfixed integerj ≥ 1 we seek asolutionθ to (4.2) of the form

θ(t,x) = eσt sin(j2x1) sin(jx2)∑

p≥1

cp sin(mpx3) (4.6)

where the sequencecp decays rapidly asp → ∞. We shall construct such a solutionθ with σ real andpositive, and in addition obtain bounds on the value ofσ. Inserting (4.6) into (4.1) and using the definitionof M3, we obtain

σp∑

p≥1

cp sin(mpx3) +∑

p≥1

cpαp

(sin(m(p + 1)x3) + sin(m(p− 1)x3)) = 0, (4.7)

where we have denoted

σp = σ + κ(j4 + j2 + (mp)2)γ (4.8)

and

αp =23Ω2(mp)2(j4 + j2 + (mp)2) + 2µ2j4

aµmj2(j4 + j2)(4.9)

for any integerp ≥ 1 (note thatj is fixed). Hereµ = β2/η. An essential feature of the coefficientsαp

is that they grow rapidly withp asp → ∞. Equation (4.9) gives the recurrence relation for the unknowncoefficientscp:

σpcp +cp+1

αp+1+cp−1

αp−1= 0, for p ≥ 2 (4.10)

σ1c1 +c2α2

= 0, for p = 1. (4.11)

Note that given anyσ > 0 (which then uniquely definesσp for all p ≥ 1 cf. (4.8)) and given anyc1 > 0,one may use the recursion relations (4.10)–(4.11) to solve for all cp with p ≥ 2. However, only for suitablevalues ofσ do thecp’s vanish sufficiently fast so thatθ isC∞ smooth. In this direction we prove:

Lemma 4.2. Let αp be defined by(4.9), where the positive integersm and j are fixed, andκ, µ,Ω, a arefixed physical parameters. Also, definec1 = α1. Then there exists a real eigenvalueσ = σ(j) > 0, andrapidly decaying sequencecpp≥2 which satisfies(4.10)–(4.11). Furthermore the lower bound

σ(j) >aµmj2(j4 + j2)

23Ω2m2(j4 + j2 + 4m2) + 2µ2j4− κ(j4 + j2 + 4m2)γ (4.12)

holds, andcp = O(Cp/(p−1)!4) asp→ ∞, for some constantC which depends onσ(j), j and the physicalparameters.

SUPERCRITICALLY DIFFUSIVE(MG) 9

Proof of Lemma 4.2.We define

ηp =cpαp−1

cp−1αp(4.13)

and write (4.10)–(4.11) in the form

σpαp + ηp+1 +1

ηp= 0, p ≥ 2 (4.14)

σ1α1 + η2 = 0, p = 1. (4.15)

Using (4.14) to writeη2 as an infinite continued fraction and equating withη2 given by (4.15) gives thecharacteristic equation

σ1α1 =1

σ2α2 − 1σ3α3− 1

σ4α4−...

. (4.16)

Recalling the definition ofσp for p ≥ 1 cf. (4.8), we observe that (4.16) is an equation with only oneunknown, namelyσ. For real values ofσ we define the infinite continued fractionFp(σ) by

Fp(σ) =1

σpαp − 1σp+1αp+1− 1

σp+2αp+2−...

(4.17)

and the function

Gp(σ) =σpαp −

√σ2pα

2p − 4

2=

2

σpαp +√σ2pα

2p − 4

(4.18)

for all p ≥ 2, and allσ such thatσ2α2 > 2. We note that due to the very rapid growth of theαp’s, for realσthe continued fractionFp defines a function which is smooth except for a set of points such withσ2α2 < 2.For the rest of the proof we will always assume thatσ is real such thatσ2α2 > 2, which also implies thatσpαp > 2 for all p ≥ 2.

We note that by constructionGp satisfies

Gp(σ) =1

σpαp −Gp(σ)=

1

σpαp − 1σpαp− 1

σpαp−...

. (4.19)

Since we haveσpαp → ∞ asp → ∞, we have thatGp(σ) → 0 asp → ∞ for every fixedσ. Also, weclearly have

G2(σ) > G3(σ) > G4(σ) > . . . ≥ 0 (4.20)

and

Gp+1(σ) < Gp(σ) < σpαp (4.21)

for all p ≥ 2. Hence, from (4.19) and (4.20) we obtain

G2(σ) >1

σ2α2 −G3(σ)>

1

σ2α2 − 1σ3α3−G4(σ)

> 0. (4.22)

An inductive argument then gives

G2(σ) >1

σ2α2 − 1σ3α3− 1

σ4α4−...

= F2(σ) ≥ 0 (4.23)

for all σ such thatσ2α2 > 2. Repeating this constructive argument we obtain thatG3(σ) > F3(σ) ≥ 0, andhence

0 < σ2α2 −G3(σ) < σ2α2 − F3(σ) < σ2α2 (4.24)

10 SUSAN FRIEDLANDER, WALTER RUSIN, AND VLAD VICOL

so that

F2(σ) =1

σ2α2 − F3(σ)>

1

α2σ2. (4.25)

In summary, we have proven that1/(σ2α2) < F2(σ) < G2(σ) for all σ such thatσ2α2 > 2, and thestraight lineσ1α1 intersects both the graph of1/(σ2α2) and the graph ofG2(σ) in this rage ofσ. Hence, itfollows that there exists a real solutionσ(j) of the equation

F2(σ(j)) = σ

(j)1 α1, (4.26)

with σ(j)2 α2 > 2, where we denoteσ(j)p = σ(j) + κ(j4 + j2 +m2p2)γ , for all p ≥ 2. That is,σ(j) is a realpositive solution of the characteristic equation (4.16). Furthermore, due to (4.23) and (4.25) after a shortcalculation we obtain an upper and a lower bound onσ(j), namely

1

α1α2< σ

(j)1 σ

(j)2 <

2

α1α2. (4.27)

For σ(j) satisfying (4.26), we now construct the sequencecp which decays rapidly asp → ∞. Therecursion relation (4.14)–(4.15) uniquely defines the values ofηp. After lettingc1 = α1, we define

cp = αpηpηp−1 . . . η2 (4.28)

for all p ≥ 2. This sequence satisfies (4.10)–(4.11) by construction. Furthermore, we observe thatηp =

−Fp(σ(j)). Repeating the arguments which gave (4.23) and (4.25) with2 replaced byp gives

−2

σ(j)p αp

< ηp <−1

σ(j)p αp

. (4.29)

Moreover, from (4.9) we have thatαp = O(p4) asp → ∞, and hence from (4.29) we obtain thatηp =O(p−4). Thus it follows from (4.28) thatcp → 0 asp → ∞, and this convergence is very fast, namelyO(Cp(p− 1)!−4) asp→ ∞, for some positive constantC = C(σ(j), µ,Ω, a,m, j). Therefore the solutionθ(j)(x, t) given by (4.6) withσ replaced byσ(j), andcp as defined by (4.28), lies in any Sobolev space, it isC∞ smooth, and even real-analytic.

We substitute forα1 andα2 from (4.9) into the bound (4.27) to obtain estimates onσ(j) given by

aµmj2(j4 + j2)

23Ω2m2(j4 + j2 + 4m2) + 2µ2j4− κ(j4 + j2 + 4m2)γ < σ(j) (4.30)

and

σ(j) <2aµmj2(j4 + j2)

23Ω2m2(j4 + j2 +m2) + 2µ2j4− κ(j4 + j2 +m2)γ . (4.31)

We recall the role of the constants in the above expressions (4.30)–(4.31) for the bounds onσ(j). Thephysical parameters areκ the coefficient of thermal diffusivity,Ω the rotation rate of the system,µ is relatedto the underlying magnetic field, anda is the magnitude of the steadybuoyancyΘ0. Expressions (4.30)and (4.31) indicate wide ranges of the physical parameters for which there exist eigenvaluesσ(j) which arepositive as postulated in the construction of the solution to (4.1). We note that in the Earth’s fluid coreκ isvery small.

Lemma 4.3. Let γ ∈ (0, 1/2). Fix a, µ,Ω, κ > 0 and an integerm ≥ 1. There exists an integerj0 = j0(a,m, µ,Ω, κ, γ) such that for allj ≥ j0 there exists aC∞ smooth initial datumθ(j)(0,x) with‖θ(j)(0, ·)‖L2 = 1 and aC∞ smooth solutionθ(j)(t,x) of the initial value problem associated with thelinearized(MG) equation(4.2), such that

‖θ(j)(t, ·)‖L2 ≥ exp(j2C∗t

)(4.32)

for all t > 0, whereC∗ = C∗(a,m, µ,Ω, κ, j0) is a positive constant defined in(4.33)below.

SUPERCRITICALLY DIFFUSIVE(MG) 11

Proof of Lemma 4.3.For anyj ≥ 1, Lemma 4.2 guarantees the existence of an eigenvalueσ(j) of the linear(MG) operatorL, with associatedC∞ smooth eigenfunctionθ(j)(0,x) given by lettingt = 0 in (4.6).Thenθ(j)(t,x) = exp(σ(j)t)θ(j)(0, ·) is a solution of (4.2). It is clear that one may normalizeθ(j)(0, ·) tohaveL2 norm equal to1, and hence‖θ(j)(t, ·)‖L2 = exp(σ(j)t).

The lemma is then proven if we pickj large enough so thatσ(j) ≥ j2C∗ for some positive constantC∗(independent ofj). This is guaranteed by the lower bound (4.30) forσ(j), if we pick j ≥ j0, wherej0 is alarge enough fixed integer such that

C∗ =aµm(j40 + j20)

23Ω2m2(j40 + j20 + 4m2) + 2µ2j40− κ

j20(j40 + j20 + 4m2)γ > 0. (4.33)

Note that whenγ ∈ (0, 1/2) such aj0 always exists, independently of the size of the physical parameters.

We now have all necessary ingredients to conclude the proof of Theorem 4.1.

Proof of Theorem 4.1.Let T > 0 andK > 0 be arbitrary, and letY ⊂ L2 be a Sobolev space. Pick anintegerm ≥ 1, and an amplitudea > 0. For these fixeda,m and physical parametersµ,Ω, κ, γ > 0, definej0 andC∗ as in Lemma 4.3. For anyj ≥ j0, Lemma 4.3 guarantees that there exists aC∞ smooth initialcondition θ(j)(0,x) which we re-normalize to have‖θ(j)(0, ·)‖Y = 1, such that the associated solutionθ(j)(t,x) of the Cauchy problem (4.2)–(4.4) satisfies

‖θ(j)(t, ·)‖L2 ≥ exp(j2C∗t)‖θ(j)(0, ·)‖L2 (4.34)

for all t > 0. We note that this solution to the linear equation is the unique3 solution inL∞(0, T ;L2). Wenow claim that there exists a sufficiently largej∗ ≥ j0 such that

‖θ(j∗)(T/2, ·)‖L2 ≥ 2K (4.35)

which would then conclude the proof of the Theorem, since‖θ(j∗)(0, ·)‖Y = 1. After a short calculation itis clear that (4.35) follows from (4.34) if we manage to provethat

exp

(j2∗C∗T

2

)‖θ(j∗)(0, ·)‖L2 ≥ 2K. (4.36)

But from the definition ofθ(j∗) cf. (4.6) above, we see that‖θ(j∗)(0, ·)‖L2 ≥ c1 = α1 ≥ 1/(Cj2∗), for somesufficiently large constantC, cf. (4.9) above. The fact that for every fixedT > 0 andC∗ > 0 the sequenceexp(j2C∗T/2)/(Cj2) diverges asj → ∞, shows that there exits some sufficiently largej∗ such that (4.36)holds, thereby concluding the proof of the theorem.

4.2. Non-linear ill-posedness in Sobolev spaces.Having established that thelinearized (MG) are ill-posed inL2, we now turn to address the Hadamard ill-posedness of thefull nonlinear(MG) equations. Theill-posedness of non-linear partial differential equations may have different sources, of varying degree ofgravity: finite time blow-up, non-uniqueness, or non-smoothness of the solution map, to name a few. As in[10] for the caseκ = 0, here we show that the fractionally diffusive(MG) equations, with0 < γ < 1/2do not posses a Lipschitz continuous solution map. Recall the definition of Lipschitz well-posedness for thenonlinear equation (see [12, Definition 1.1], or [10, Definition 2.9]):

Definition 4.4 (Lipschitz local well-posedness).Let Y ⊂ X ⊂ W 1,4 be Banach spaces. The Cauchyproblem for the(MG) equation

∂tΘ+U · ∇Θ+ κ(−∆)γΘ = S (4.37)

∇ ·U = 0, Uj =MjΘ (4.38)

3The proof of uniqueness of finite energy solutions to the linear equation follows from a representation of the solution asaFourier series (see [10], Proposition 2.8).

12 SUSAN FRIEDLANDER, WALTER RUSIN, AND VLAD VICOL

is locally Lipschitz(X,Y ) well-posed, if there exist continuous functionsT,K : [0,∞)2 → (0,∞), the timeof existence and the Lipschitz constant, so that for every pair of initial data Θ(1)(0, ·),Θ(2)(0, ·) ∈ Y thereexist unique solutionsΘ(1),Θ(2) ∈ L∞(0, T ;X) of the initial value problem associated to(4.37)–(4.38),that additionally satisfy

‖Θ(1)(t, ·)−Θ(2)(t, ·)‖X ≤ K‖Θ(1)(0, ·) −Θ(2)(0, ·)‖Y (4.39)

for everyt ∈ [0, T ], whereT = T (‖Θ(1)(0, ·)‖Y , ‖Θ(2)(0, ·)‖Y ) andK = K(‖Θ(1)(0, ·)‖Y , ‖Θ(2)(0, ·)‖Y ).Remark 4.5. Clearly the time of existenceT and the Lipschitz constantK also depend on‖S‖L∞(0,∞;Y ),and onκ, but we have omitted this dependence in Definition 4.4 since it is the same for both solutionsΘ(1)

andΘ(2).

Remark 4.6. If Θ(2)(t, ·) = 0 andX = Y , Definition 4.4 recovers the usual definition of local well-posedness with a continuous solution map. However, Defintion 4.4 allows the solution map to lose regularity,which is usually needed in order to obtain Lipschitz continuity of the solution map for first order quasi-linearequations. Hence, the typical pairs of spaces(X,Y ) that we have in mind here areX = Hs, andY = Hs+1,with s > 1 + 3/4.

The main result of this section is the following theorem.

Theorem 4.7(Nonlinear ill-posedness in Sobolev spaces).The(MG) equations(4.37)–(4.38), with γ ∈(0, 1/2) are locally Lipschitz(X,Y ) ill-posed in Sobolev spacesY ⊂ X embedded inW 1,4(T3), in thesense of Definition 4.4.

For the purpose of our ill-posedness result, we shall letΘ(2)(t,x) be the steady stateΘ(x3) intro-duced earlier in (4.4). We considerX to be a Sobolev space with high enough regularity so that∂tΘ ∈L∞(0, T ;L2), which implies thatΘ is weakly continuous on[0, T ] with values inX, making sense of theinitial value problem associated to (4.37)–(4.38). The proof of Theorem 4.7 follows from the stronglinearill-posedness obtained in Theorem 4.1 and a fairly generic perturbative argument (cf. [20, Thorem 2] or [24,pp. 183]). The proof of Theorem 4.7 follows the lines of the proof for the non diffusive problem given in[10].

Proof of Theorem 4.7.Since the Sobolev spaceX embeds inH1, andγ ∈ (0, 1/2), the linearized operatorLΘ = −M3Θ ∂3Θ0 − κ(−∆)γΘ mapsX continuously intoL2, and sinceX ⊂ W 1,4, the nonlinearityNΘ = −MjΘ ∂jΘ is bounded as‖NΘ‖L2 ≤ ‖∇Θ‖2L4 ≤ C‖Θ‖2X , for some constantC > 0.

Fix the steady stateΘ0(x3) ∈ Y , as given by (4.4). Also, fix a smooth functionψ0 ∈ Y , normal-ized to have‖ψ0‖Y = 1, to be chosen precisely later. The proof is by contradiction. Assume that theCauchy problem for the(MG) equation (4.37)–(4.38) is Lipschitz locally well-posed in(X,Y ). ConsiderΘ(2)(0,x) = Θ0(x3), so thatΘ(2)(t,x) = Θ0(x3) for anyt > 0. Also let

Θǫ(0,x) = Θ0(x3) + ǫψ0(x),

for every0 < ǫ < ‖Θ0‖Y . To simplify notation we writeΘǫ instead ofΘ(1,ǫ). By Definition 4.4, forevery ǫ as before there exists a positive timeT = T (‖Θ0‖Y , ‖Θǫ‖Y ) and a positive Lipschitz constantK = (‖Θ0‖Y , ‖Θǫ‖Y ) such that by (4.39) and the choice ofψ0 we have

‖Θǫ(t, ·)−Θ0(·)‖X ≤ Kǫ (4.40)

for all t ∈ [0, T ]. We note that since‖Θǫ(0, ·)‖Y ≤ ‖Θ0‖Y + ǫ ≤ 2‖Θ0‖Y , due to the continuity ofT andK with respect to the second coordinate, we may chooseK = K(‖Θ0‖Y ) > 0 andT = T (‖Θ0‖Y ) > 0independent ofǫ ∈ (0, ‖Θ0‖Y ), such that (4.40) holds on[0, T ].

Writing Θǫ as anO(ε) perturbation ofΘ0, we define

ψǫ(t,x) =Θǫ(t,x)−Θ0(x3)

ε, (4.41)

SUPERCRITICALLY DIFFUSIVE(MG) 13

for all t ∈ [0, T ] and allǫ as before. It follows from (4.40) thatψǫ is uniformly bounded with respect toǫin L∞(0, T ;X). Therefore, there exists a functionψ, the weak-∗ limit of ψǫ in L∞(0, T ;X). Note thatψǫ

solves the Cauchy problem

∂tψǫ = Lψǫ + ǫN(ψǫ) (4.42)

ψǫ(0, ·) = ψ0. (4.43)

Due to the choice ofX, we have the bound

‖Nψǫ‖L2 ≤ C‖ψǫ‖2X ≤ CK2, (4.44)

and from (4.42) we obtain that∂tψǫ is uniformly bounded with respect toǫ in L∞(0, T ;L2). Therefore theconvergenceψǫ → ψ is strong when measured inL2. Sendingǫ to 0 in (4.42), and using (4.44), it followsthat

∂tψ = Lψ

ψ(0, ·) = ψ0

holds inL∞(0, T ;L2), and this solution is unique since the problem is now linear.In addition, the solutionψ inherits from (4.40) the upper bound

‖ψ(t, ·)‖L2 ≤ K (4.45)

for all t ∈ [0, T ]. But this is a contradiction with Theorem 4.1. Due to the existence of eigenfunctions forthe linearized operator with arbitrarily large eigenvalues, one may chooseψ0 (as in Lemma 4.3) to yield alarge enough eigenvalue so that in timeT/2 the solution grows to haveL2 norm larger than2K (similarlyto the proof of Theorem 4.1), therefore contradicting (4.45).

5. THE REGIME γ = 1/2: A DICHOTOMY IN TERMS OF THE SIZE OF THE DATA

If the initial data is small with respect toκ, then one may use energy estimates to show that there exists aunique global smooth solution evolving from this data (cf. Section 5.2 below). However, the proof does notapply for the case of large data, not even to prove the local existence of solutions. In the case of large data,we may construct a steady state such that solutions evolvingfrom initial data which is arbitrarily close tothis steady state diverge from it at an arbitrarily large rate, for positive time (cf. Section 5.1 below), i.e. theequations are Hadamard ill-posed.

5.1. Ill-posedness forγ = 1/2 and large data. In Section 4 above we have shown that forγ < 1/2 the(MG) equations are Hadamard ill-posed in Sobolev spaces, in the sense that there is no Lipschitz continuoussolution map (see Theorem 4.7). The main ingredient in the proof of ill-posedness forγ ∈ (0, 1/2) was toshow that the linearized(MG) operator

Lθ = −M3θ∂3Θ0 − κ(−∆)γθ (5.1)

where

Θ0 = a sin(mx3) (5.2)

is a steady state associated to the source termS = κam2γ sin(mx3), has a sequence of eigenvaluesσ(j)

which diverge to infinity asj → ∞ (see Lemmas 4.2 and 4.3).We emphasize that it isonly in the proof of Lemma 4.3 whereγ < 1/2, rather thanγ ≤ 1/2 is used.

Indeed, in order to prove that givenanyreala > 0 and any integerm ≥ 1 there exists some sufficiently largeintegerj0 such that the constantC∗ of (4.33) is strictly positive,γ < 1/2 is both necessary and sufficient.In the caseγ = 1/2, we can prove the following alternative to Lemma 4.3, which states that only ifa issufficiently large with respect toκ (in terms ofµ andΩ), then one obtains a sequence of eigenvalues whichdiverge to∞. Namely:

14 SUSAN FRIEDLANDER, WALTER RUSIN, AND VLAD VICOL

Lemma 5.1. Letγ = 1/2. Fix a, µ,Ω, andm. For values ofa such that

κ <aµm

25Ω2m2 + 2µ2(5.3)

and all integersj ≥ m, the statement of Lemma 4.3 holds with the constantC∗ given by

C∗ =aµm

25Ω2m2 + 2µ2− κ. (5.4)

Proof of Lemma 5.1.The proof follows from the lower bound (4.30) on the eigenvalueσ(j), similarly to theproof of Lemma 4.3. The role of condition (5.3) is to guarantee that there exists a large enoughj0 such that

aµm(j40 + j20)

23Ω2m2(j40 + j20 + 4m2) + 2µ2j40− κ

j20(j40 + j20 + 4m2)γ > 0.

To avoid repetition we omit further details.

For those values ofa for which (5.3) holds, Lemma 5.1 shows that the operatorL has unbounded spectrumin the unstable region, and hence one may prove with virtually no modificationsfrom theγ ∈ (0, 1/2) case,that the equations are ill-posed. More precisely we have

Theorem 5.2(Ill-posedness for large data).LetY ⊂ X ⊂W 1,4(T3) be Sobolev spaces, and letγ = 1/2.Givenκ, µ,Ω > 0, fix an integerm ≥ 1. Let a > 0 be sufficiently large such that(5.3) holds and letΘ

(1)0 = a sin(mx3) ∈ Y be a steady state of(1.1). Then, given anyT > 0 and anyK > 0, there exists

an initial conditionΘ(2)0 ∈ Y and a corresponding solutionΘ(2) ∈ L∞(0, T ;X) of the Cauchy problem

(1.1)–(1.3), such that

supt∈[0,T ]

‖Θ(2)(t, ·) −Θ(1)(t, ·)‖X ≥ 2K‖Θ(2)0 −Θ

(1)0 ‖Y (5.5)

whereΘ(1)(t, ·) = Θ(1)0 . That is, the(MG) equations are locally Lipschitz (X,Y) ill-posed when the data is

large.

The proof of Theorem 5.2 is the same as the proof of Theorem 4.7above. The only difference in the caseγ = 1/2 is to use Lemma 5.1 to show that the linear equations have unbounded unstable spectrum. We omitdetails.

5.2. Well-posedness forγ = 1/2 and small data. As shown in the previous section, we can exhibit initialdata, for which the system (1.1)–(1.2) is ill-posed in the above described sense. Note also, that the proof ofTheorem 3.1, fails for the valueγ = 1/2. Thus,γ = 1/2 indeed is the limit of the local well-posednesstheory. Nonetheless, we still can prove that the consideredsystem is globally well-posed for small data.

Theorem 5.3(Global existence for small data).Lets > 5/2 and assume that the initial dataΘ0 ∈ Hs(T3)and S ∈ L2(0,∞;Hs−γ(T3)) have zero-mean onT3. There exists a sufficiently small constantε > 0depending onκ, such that if‖Θ0‖αL2‖Θ0‖1−α

Hs + ‖S‖L∞(0,∞;Hs−γ) ≤ ε, whereα = 1 − (5/2)/s, then theunique smooth solution

Θ ∈ L∞(0, T ;Hs(T3)) (5.6)

of the Cauchy problem(1.1)–(1.6) is global in time.

Proof. We proceed as in the proof of Theorem 3.2 and obtain the energyestimate

1

2

d

dt‖ΛsΘ‖2L2 + κ‖Λs+ 1

2Θ‖2L2 ≤ ‖Λs− 12S‖L2‖Λs+ 1

2Θ‖L2 +

∣∣∣∣∫

R3

Λs− 12 (U · ∇Θ)Λs+ 1

2Θ

∣∣∣∣ . (5.7)

The second term on the right side is estimated using the product estimate in Proposition 2.1. Thus we obtain,similarly to (3.8)

‖Λs− 12 (U · ∇Θ)‖L2 ≤ C(‖∇Θ‖L∞ + ‖U‖L∞)‖Λs+ 1

2Θ‖L2 . (5.8)

SUPERCRITICALLY DIFFUSIVE(MG) 15

Sinces > 5/2, we get

‖U‖L∞ + ‖∇Θ‖L∞ ≤ C‖Θ‖αL2‖ΛsΘ‖1−αL2 , (5.9)

whereα = 1− (n/2 + 1)/s. Combining estimates (5.7)-(5.9) gives

1

2

d

dt‖ΛsΘ‖2L2 + κ‖Λs+ 1

2Θ‖2L2 ≤ 1

2κ‖Λs− 1

2S‖2L2 +κ

2‖Λs+ 1

2Θ‖2L2 + C‖Θ‖αL2‖ΛsΘ‖1−αL2 ‖Λs+ 1

2Θ‖2L2 ,

(5.10)

which in turn leads to1

2

d

dt‖ΛsΘ‖2L2 +

κ

2‖Λs+ 1

2Θ‖2L2 ≤ 1

2κ‖Λs− 1

2S‖2L2 + C‖Θ‖αL2‖ΛsΘ‖1−αL2 ‖Λs+ 1

2Θ‖2L2 . (5.11)

We obtain the desired result as in the proof of Theorem 3.2.

Remark 5.4. Note that conditions in Theorem 3.2 are consistent with the above theorem if we setγ = 1/2.

Remark 5.5. We note that the ill-posedness result in the caseγ = 1/2 requires the constant defined in(5.4) to be positive. This does not hold when the value ofa is sufficiently small (depending onκ, µ, andµ) no matter what value ofm ≥ 1 is picked. Recalling thata measures the magnitude of the initial dataassociated with‖Θ0‖L2 , we observe that the well-posedness result whenγ = 1/2 is only proven in the caseof small data and small force (see Theorem 5.3), and hence fora sufficiently small. Therefore our large dataill-posedness result is consistent with the small-data well-posedness result whenγ = 1/2.

6. IMPROVEMENT IN REGULARITY FOR“ WELL-PREPARED” DATA AND SOURCE

In this section we explore the following observation: if thefrequency support ofΘ lies on a suitableplane, then the operatorM is “mild” when it acts onΘ, i.e. it behaves like an order0 operator, and hencethe corresponding velocityU is as smooth asΘ. This enables us to obtain improved well-posedness resultsover the generic setting when no conditions on the Fourier spectrum of the initial data (and source term) areimposed. For instance the local existence an uniqueness of smooth solutions holds even if0 < γ < 1/2, asetting in which we know that for generic initial data the problem is ill-posed.

To be more precise letq ∈ Q be a rational number4 with q 6= 0. We define the frequency planePq to bethe set

Pq = k = (k1, k2, k3) ∈ Z3 : k2 = qk1. (6.1)

We shall need the following straightforward observation:

Proposition 6.1. Assumef, g are smoothT3 periodic functions, with frequency supportsupp(f), supp(g) ⊂Pq for someq ∈ Q \ 0. Then we havesupp(f ± g), supp(f g), supp(Mjf) ⊂ Pq, for all j ∈ 1, 2, 3.

Proof of Propostion 6.1.Clearly supp(f + g) = supp(f + g) ⊂ supp(f) ∪ supp(g) ⊂ Pq. To see thatthat the frequency support of the functionMjf lies onPq, we just note thatsupp(Mjg) = supp(Mj f) ⊂supp(Mj) ∩ supp(f) ⊂ Pq. To analyze the frequency support offg = f ∗ g, note thatsupp(f ∗ g) ⊂supp(f) + supp(g) ⊂ Pq, sincePq is closed under addition of vectors, concluding the proof.

Lemma 6.2. If f is a smoothT3 periodic function withf(k1, k2, 0) = 0 for all k1, k2 ∈ Z, and such thatsupp(f) ⊂ Pq for someq ∈ Q \ 0, then there exists a universal constantC∗ = C∗(q,Ω, β, η) > 0 suchthat ∣∣∣(Mjf)(k)

∣∣∣ =∣∣∣Mj(k)f(k)

∣∣∣ ≤ C∗|f(k)| (6.2)

for all k ∈ Z3, and for allj ∈ 1, 2, 3. Additionally, the constantC∗ blows-up asq → 0.

4Since we are in the periodic setting when the frequency is supported onZ3.

16 SUSAN FRIEDLANDER, WALTER RUSIN, AND VLAD VICOL

Proof of Lemma 6.2.It is clear that (6.2) has to be proven only fork such thatk3 6= 0, since otherwise wehave thatf(k) = 0 and the statement holds trivially. We now consider each of the casesj ∈ 1, 2, 3.Without loss of generality takeq > 0.

For j = 1, a short algebraic computations gives∣∣∣M1(k)

∣∣∣ ≤(2Ω(2q + q3) + q2β2/η

) k31k3 + k1k33

2Ω2k43 + q4β4/η2k41(6.3)

from which it follows that|M1(k)| ≤ C, for a suitable constantC, by using the inequalityk31k3 + k1k33 ≤

k41 + k43. Similarly to (6.3) it follows that fork ∈ Pq we have∣∣∣M2(k)

∣∣∣ ≤(2Ω(2 + q2) + q3β2/η

) k31k3 + k1k33

2Ω2k43 + q4β4/η2k41(6.4)

and∣∣∣M3(k)

∣∣∣ ≤(q2(1 + q2)β2/η

) k412Ω2k43 + q4β4/η2k41

(6.5)

which concludes the proof of the lemma upon usingk31k3 + k1k33 ≤ k41 + k43 .

6.1. Local existence and uniqueness for0 < γ < 1. The main result of this section is:

Theorem 6.3(Local existence).Let γ ∈ (0, 1), and fixs > 5/2 − γ. Assume thatΘ0 ∈ Hs(T3) andS ∈ L∞(0,∞;Hs−γ(T3)) have zero-mean onT3 and are such that

supp(Θ0) ⊂ Pq and supp(S(t, ·)) ⊂ Pq, (6.6)

for some fixedq ∈ Q \ 0 and all t ≥ 0. Then there exists aT > 0 and a unique smooth solution

Θ ∈ L∞(0, T ;Hs(T3)) ∩ L2(0, T ;Hs+γ(T3)) (6.7)

of the Cauchy problem(1.10)–(1.11), such that

supp(Θ(t, ·)) ⊂ Pq (6.8)

for all t ∈ [0, T ).

Proof of Theorem 6.3.In order to construct the local in time solutionΘ, with frequency support onPq,consider the sequence of approximationsΘnn≥1 given by the solutions of

∂tΘ1 + (−∆)γΘ1 = S (6.9)

Θ1(0, ·) = Θ0 (6.10)

and respectively

∂tΘn +Un−1 · ∇Θn + (−∆)γΘn = S (6.11)

Un−1 = MΘn−1 (6.12)

Θn(0, ·) = Θ0 (6.13)

for all n ≥ 2. One may solve (6.9)–(6.10) explicitly using Duhamel’s formula, and noting that the operatorexp((−∆)γt) is a Fourier multiplier with non-zero symbol given byexp(|k|γt), it follows from Proposi-tion 6.1 thatsupp(Θ1(t, ·)) ⊂ Pq for all t ≥ 0. In addition, we have the bound

‖ΛsΘ1‖2L∞(0,T ;L2) + ‖Λs+γΘ1‖2L2(0,T ;L2) ≤ ‖ΛsΘ0‖2L2 + T‖S‖2L∞(0,T ;Hs−γ) (6.14)

for all T > 0. On the other hand, in order to solve (6.11)–(6.13) we appealto Theorem A.1. Indeed,by the inductive assumption we have thatΘn−1 ∈ L∞(0, T ;Hs) and alsosupp(Θn−1(T, ·)) ⊂ Pq forall T > 0. Hence, by applying Lemma 6.2 we haveUn−1 ∈ L∞(0, T ;Hs) and by Proposition 6.1 wehavesupp(Un−1(t, ·)) for all t > 0. Therefore all the conditions of Theorem A.1 are satisfied, by letting

SUPERCRITICALLY DIFFUSIVE(MG) 17

v = Un−1, and there hence exists a unique solutionΘn ∈ L∞(0, T ;Hs)∩L2(0, T ;Hs+γ) of (6.11)–(6.13),such thatsupp(Θn(t, ·)) ⊂ Pq for all t ∈ [0, T ).

To prove that the sequenceΘn converges, we first prove that it is bounded. Fix a timeT to be chosenlater such thatT < ‖ΛsΘ0‖2L2/‖S‖2L∞(0,T ;Hs−γ). It hence follows from (6.14) that

‖ΛsΘj‖2L∞(0,T ;L2) + ‖Λs+γΘj‖2L2(0,T ;L2) ≤ 2‖ΛsΘ0‖2L2 (6.15)

holds whenj = 1. We assume inductively that (6.15) holds for all1 ≤ j ≤ n− 1, and proceed to prove thatit holds for j = n. SinceUn−1 is obtained fromΘn−1 by a bounded Fourier multiplier (cf. Lemma 6.2),there exists a positive constantCq, depending onq (and on the physical parametersΩ, β, η), such that

‖ΛsUn−1(t, ·)‖2L2 ≤ Cq‖ΛsΘn−1(t, ·)‖2L2 (6.16)

for all t ≥ 0. ApplyingΛs to (6.11) and taking anL2 inner product withΛsΘn we hence obtain

1

2

d

dt‖ΛsΘn‖2L2 + ‖Λs+γΘn‖2L2

≤ ‖ΛsΘn‖L2‖[Un−1 · ∇,Λs]Θn‖L2 + ‖Λs+γΘn‖L2‖Λs−γS‖L2

≤ C‖ΛsΘn‖L2 (‖∇Un−1‖L3/γ‖ΛsΘn‖L6/(6−γ) + ‖ΛsUn−1‖L2‖∇Θn‖L∞) + ‖Λs+γΘn‖L2‖Λs−γS‖L2

≤ C‖ΛsΘn‖L2‖ΛsUn−1‖L2‖Λs+γΘn‖L2 + ‖Λs+γΘn‖L2‖Λs−γS‖L2

≤ 1

2‖Λs+γΘn‖2L2 + Cs,γ‖Λs

Un−1‖2L2‖ΛsΘn‖2L2 + ‖Λs−γS‖2L2 (6.17)

whereCs,γ > 0 is a constant. Above we have used the fact that∇ · Un−1 = 0 in order to write thecommutator, and also the product and commutator estimates of Proposition 2.1. Using the bound (6.16),and the inductive assumption (6.15), it follows from (6.17)that

d

dt‖ΛsΘn‖2L2 + ‖Λs+γΘn‖2L2 ≤ 4Cs,γCq‖ΛsΘ0‖2L2‖ΛsΘn‖2L2 + 2‖Λs−γS‖2L2 (6.18)

and hence

‖ΛsΘn(t, ·)‖2L2 +

∫ t

0‖Λs+γΘn(τ, ·)‖2L2dτ

≤ ‖ΛsΘ0‖2L2e4Cs,γCq‖ΛsΘ0‖2

L2 t +

∫ t

0e4Cs,γCq‖ΛsΘ0‖2

L2 (t−τ)‖Λs−γS(τ, ·)‖2L2dτ

≤ e4Cs,γCq‖ΛsΘ0‖2L2 t

(‖ΛsΘ0‖2L2 + t‖S‖2L∞(0,∞;Hs−γ)

)(6.19)

for all t ∈ (0, T ). Therefore, letting

T ≤ min

‖ΛsΘ0‖2L2

2‖S‖2L∞(0,∞;Hs−γ)

,ln 4/3

4Cs,γCq‖ΛsΘ0‖2L2

(6.20)

we obtain that from (6.19) that (6.15) holds forj = n, and so by induction it holds for allj ≥ 1. This showsthat the sequenceΘn is uniformly bounded inL∞(0, T ;Hs) ∩ L2(0, T ;Hs+γ).

Moreover, we may show that the sequenceΘn is Cauchy inL∞(0, T ;Hs−1+γ). To see this, considerthe difference of two iteratesΘn := Θn −Θn−1. It follows from (6.11) thatΘn is a solution of

∂tΘn + (−∆)γΘn +Un−1 · ∇Θn + Un−1 · ∇Θn−1 = 0 (6.21)

Θn(0, ·) = 0 (6.22)

18 SUSAN FRIEDLANDER, WALTER RUSIN, AND VLAD VICOL

for all n ≥ 2, where we also denotedUn = MΘn. ApplyingΛs−1+γ to (6.21), taking anL2 inner productwith Λs−1+γΘn, using∇ ·Un−1 = 0, and the calculus inequalities of Proposition 2.1, we obtain

1

2

d

dt‖Λs−1+γΘn‖2L2 + ‖Λs−1+2γΘn‖2L2

≤ ‖Λs−1+γΘn‖L2‖[Un−1 · ∇,Λs−1+γ ]Θn‖L2 + ‖Λs−1+γΘn‖L2‖Λs−1+γ(Un−1 · ∇Θn)‖L2

≤ Cs‖Λs−1+γΘn‖L2

(‖∇Un−1‖L∞‖Λs−1+γΘn‖L2 + ‖Λs−1+γ

Un−1‖L6‖∇Θn‖L3

)

+ Cs‖Λs−1+γΘn‖L2

(‖Un−1‖L∞‖Λs+γΘn‖L2 + ‖Λs−1+γ

Un−1‖L2‖∇Θn‖L∞

)

≤ Cs,q‖Λs−1+γΘn‖L2

(‖Λs+γΘn−1‖L2‖Λs−1+γΘn‖L2 + ‖Λs−1+γΘn−1‖L2‖Λs+γΘn‖L2

).

(6.23)

In the last inequality above we have used the assumptions > 5/2 − γ, and the estimate (6.16). It hencefollows from (6.23), (6.15), and the Cauchy-Schwartz inequality that

supt∈[0,T ]

‖Λs−1+γΘn(t, ·)‖L2 ≤ supt∈[0,T ]

‖Λs−1+γΘn−1(t, ·)‖L2

∫ T

0‖Λs+γΘn(t, ·)‖L2e(

∫ Tt ‖Λs+γΘn−1(τ,·)‖L2dτ)dt

≤ supt∈[0,T ]

‖Λs−1+γΘn−1(t, ·)‖L2

√2TCs,q‖ΛsΘ0‖L2e

√2TCs,q‖ΛsΘ0‖L2dt.

(6.24)

Recalling (6.20), if we letT be such that

T = min

‖ΛsΘ0‖2L2

2‖S‖2L∞(0,∞;Hs−γ)

,ln 4/3

4Cs,γCq‖ΛsΘ0‖2L2

,C2s,q

8 exp(√

2 ln 4/3Cs,q/√Cs,γCq)

(6.25)

we obtain from (6.23) that

supt∈[0,T ]

‖Λs−1+γΘn(t, ·)‖L2 ≤ 1

2sup

t∈[0,T ]‖Λs−1+γΘn−1(t, ·)‖L2 . (6.26)

ThusΘn is Cauchy inL∞(0, T ;Hs−1+γ), and henceΘn converges strongly toΘ in L∞(0, T ;Hs−1+γ).Noting that s − 1 + γ > 3/2, this shows that the strong convergence occurs in a Holder space rela-tive to x, which is sufficient to prove that the limiting functionΘ ∈ L∞(0, T ;Hs) ∩ L2(0, T ;Hs+γ) ∩Lip(0, T ;Hmin(s−2γ,s−1)) is a solution of the initial value problem (1.1)–(1.2).

To conclude the proof of the theorem, we note that ifΘ(1) andΘ(2) are two solutions of (1.1)–(1.2), thenΘ = Θ(1) −Θ(2) solves

∂tΘ+ (−∆)γΘ+U(1) · ∇Θ+U · ∇Θ(2) = 0 (6.27)

with initial conditionΘ(0, ·) = 0. An L2 estimate on (6.27) shows thatΘ(t, ·) = 0 for all t ∈ [0, t), sinceΘ(2) ∈ L∞(0, T ;Hs) and since ifΘ(1) andΘ(2) have frequency support onPq, so doesΘ.

Corollary 6.4 (Local existence for well-prepared data).Let j1, j2 ∈ Z \ 0, and fix s, γ as in thestatement of Theorem 6.3. Assume thatΘ0 ∈ Hs(T3) is such thatΘ0(k1, k2, k3) 6= 0 if and only if(k1, k2) = ±(j1, j2) . Similarly, assume that forcingS ∈ L∞(0, Ts;H

s−γ) is such thatS(k1, k2, k3, t) 6= 0if and only if (k1, k2) = ±(j1, j2), for all t ∈ [0, Ts). Then there existsT ∈ (0, Ts] and a unique solutionΘ ∈ L∞(0, T ;Hs) ∩ L2(0, T ;Hs+γ) of the initial value problem(1.1)–(1.2), and in addition we have thatΘ(k1, k2, k3, t) = 0 wheneverk2/k1 6= j2/j1.

Proof of Corollary 6.4.Let q = j2/j1 ∈ Q. The conditions on the initial data and the force imply inparticular that their frequency support likes on the planePq. The existence of a unique smooth solutionΘ,with frequency support lying onPq follows directly from Theorem 6.3. But(k1, k2, k3) ∈ Pq is equivalentto k2 = qk1 = j2k1/j1, so thatΘ(k1, k2, k3) 6= 0 only if k2/k1 = j2/j1.

SUPERCRITICALLY DIFFUSIVE(MG) 19

6.2. Global existence and uniqueness for well-prepared data.In the previous section we have proventhat forγ ∈ (0, 1), given initial dataΘ0 and source termS which are well-prepared, i.e. they have Fouriersupport on a planePq for someq > 0, then there exists a local in time solutionΘ of the Cauchy problem(1.1)–(1.3), which has the property that its Fourier support also lies onPq. In this section we show that forγ ∈ [1/2, 1) the local in time solution may be continued for all time, while in the caseγ ∈ (0, 1/2) the sameresult holds but under the additional assumption that the initial data is small with respect toκ.

Theorem 6.5 (γ ≥ 1/2: Global existence for large data). Let s,Θ0 and S be as in the statement ofTheorem 6.3. Ifγ ≥ 1/2, the unique smooth solutionΘ of the Cauchy problem(1.10)–(1.11) is global intime.

Proof of Theorem 6.5.Given the conditions on the initial data and of the source term, and using the prop-erties exhibited in Proposition 6.1, the solution constructed in Theorem 6.3 satisfiessupp(Θ) ⊂ Pq.By Lemma 6.2 we have thatU is obtained fromΘ by a boundedFourier multiplier, and hence by theHormander-Mikhlin theorem we have

‖U‖W s,p ≤ C‖Θ‖W s,p (6.28)

for all 2 ≤ p < ∞, and alls ≥ 0. Therefore the regimeγ > 1/2 becomes “sub-critical” for such solutions,since the mapΘ 7→ U is bounded in Sobolev spaces. Therefore, energy estimates which are similar tothose used to prove the global regularity of the sub-critical SQG equation (cf. [5]), may be used with minormodifications to prove Theorem 6.5 in the settingγ > 1/2.

The caseγ = 1/2 is slightly more delicate. In the case of the critical (SQG) equation, global regularity inthe critical case has only been established recently (cf. [2, 16] ). Due to the anisotropic nature of the symbolM , it turns out that it is slightly easier to see that the DeGiorgi-inspired proof of [2] also takes care of theγ = 1/2 case of this theorem. The only fact we must verify is that whenΘ ∈ L∞, andsupp(Θ) ∈ Pq,then5

U = M [Θ] ∈ BMO. Once we prove this, the DeGiorgi iteration scheme of [2], shows thatΘ(t, ·)is Holder continuous fort > 0, and one may use energy arguments to conclude the proof of Theorem 6.5,similarly to [10, Appendix]. Lastly, to verify thatU ∈ BMO, define a new operatorM q as the Fouriermultiplier with symbolM (k)ϕq(k), whereϕq(k) is a function which is identically1 onPq, and vanishesidentically at a fixed distance away fromPq. The advantage is thatM q is now a periodic pseudo-differentialoperator of order0, and hence mapsL∞ to periodicBMO (cf. [17]). This is attributed to boundedness ofM q, inherited from the symbolsM j , and follows from the fact thatPq and its collar neighborhood arenot entirely contained in the curved region of the frequencyspace whereMj become unbounded. Lastlyone may verify that whensupp(Θ) ⊂ Pq, thenM q[Θ] = M [Θ], thereby concluding the proof of thetheorem.

Theorem 6.6(γ < 1/2: Global existence for small data). Let γ, s,Θ0 andS be as in the statement ofTheorem 6.3. Ifγ < 1/2, there exists a sufficiently small constantε > 0, such that if‖Θ0‖αL2‖Θ0‖1−α

Hs +

‖Θ0‖αL2‖S‖1−αL∞(0,∞;Hs−γ)

≤ ε, whereα = 1 − (5/2 − γ)/s, then the unique smooth solutionΘ of theCauchy problem(1.10)–(1.11)is global in time.

Proof of Theorem 6.6.SinceM acts as an operator of order0 when applied to functions with frequencysupport onPq, from the point of view of energy estimates the situation we are in is exactly the same as for thesuper-critical (SQG) equation. The proof the theorem follows from the same arguments used in [6, 14, 26]to show the global well-posedness of small solutions to the super-critical (SQG) equation.

5Note that for genericΘ ∈ L∞ functionsM [Θ] does not belong toBMO, but rather toBMO−1 (cf. [9, Section 4]).

20 SUSAN FRIEDLANDER, WALTER RUSIN, AND VLAD VICOL

APPENDIX A. SMOOTH SOLUTIONS TO A LINEAR EQUATION

FOR DATA WITH “ THIN” FOURIER SUPPORT

The goal of this appendix is to prove the existence of smooth solutions to the scalarlinear equation

∂tθ + v · ∇θ + (−∆)γθ = S (A.1)

θ|t=0 = θ0 (A.2)

where the initial datumθ0, thegivendivergence-free drift velocity fieldv, and the external source termS,all have frequency support in the same planePq, i.e.

supp(θ0), supp(S(t, ·)), supp(v(t, ·)) ⊂ Pq (A.3)

for all t ∈ [0, T ], for someq ∈ Q, q 6= 0. The main result is:

Theorem A.1 (Existence of solutions with support on a plane).Let γ ∈ (0, 1), d ≥ 2, and fixs >d/2+1−γ. Givenθ0 ∈ Hs(Td), a divergence-freev ∈ L∞(0, T ;Hs(Td)), andS ∈ L∞(0, T ;Hs−γ(Td)),such that(A.3) holds, there exists a unique solution

θ ∈ L∞(0, T ;Hs(Td)) ∩ L2(0, T ;Hs+γ(Td))

of the initial value problem(A.1)–(A.2), and we have that

supp(θ(t, ·)) ⊂ Pq

holds for allt ∈ [0, T ].

Remark A.2. The main difficulty in proving Theorem A.1 is in designing an iteration scheme which is bothsuitable for energy estimates, and preserves the feature that in each iteration step the frequency support ofthe approximation lies onPq. In this direction we note that a scheme such that∂tθn+1 is given in termsof v · ∇θn automatically preserves the frequency support onPq in view of Lemma 6.1, but is not suitablefor closing the estimates at the level of Sobolev spaces. On the other hand, if we consider∂tθ(n+1) todepend onv · ∇θn+1, while energy estimates are now clear, it seems difficult to inductively obtain thatsupp(θn+1) ⊂ Pq.

Proof of Theorem A.1.Since the iteration scheme which is suitable for controlling the frequency support ofthe solution is “loosing” a derivative, we regularize (A.1)–(A.2) with hyper-dissipation as

∂tθε + v · ∇θε + (−∆)γθε − ε∆θε = S (A.4)

θε(0, ·) = θ0 (A.5)

for ε ∈ (0, 1], and later pass to the limitε → 0 in order to obtain a solution of the original system. SincevandS are smooth, andv is divergence-free, it follows from our earlier paper [9] that there exists a uniqueglobal (or as long asv andS permit) smooth solutionθε of (A.4)–(A.5), with

θε ∈ L∞(0, T ;Hs) ∩ L2(0, T ;Hs+γ) ∩ εL2(0, T ;Hs+1). (A.6)

and the solution is bounded in the above spacesindependently ofε ∈ (0, 1].The advantage of considering the system (A.4)–(A.5) over (A.1)–(A.2), is that for the hyper-dissipative

system, we canconstructa smooth solution (as will be shown below), which has frequency support lyingin Pq. Since (A.4) is linear, and we work in the smooth category, i.e. s > d/2 + 1 − γ, the uniquesmooth solution of (A.4)–(A.5) satisfiessupp(θε(t, ·)) ⊂ Pq for all t ∈ [0, T ). We note already that theε-independent bounds in the regularity class (A.6), will allow us to pass to a limitθε → θ, asε → 0, andthis limiting function will automatically satisfysupp(θ) ⊂ Pq, since the latter space is closed and discrete.

We now proceed to construct a solutionθε of (A.4)–(A.5), which has the desired frequency supportproperty. We consider the following iterative scheme: the first iterate is given by the solution of

∂tθε1 + (−∆)γθε1 − ε∆θε1 = S (A.7)

θε1(0, ·) = θ0 (A.8)

SUPERCRITICALLY DIFFUSIVE(MG) 21

for all t ∈ [0, T ), while then+ 1st iterate is given as the unique solution of

∂tθεn+1 + v · ∇θεn + (−∆)γθεn+1 − ε∆θεn+1 = S (A.9)

θεn+1(0, ·) = θ0 (A.10)

for all n ≥ 1. We note that the solutions of (A.7)–(A.8) and (A.9)–(A.10)respectively, may be writtenexplicitly using the Duhamel formula

θε1(t) = e(ε(−∆)+(−∆)γ )tθ0 +

∫ t

0e(ε(−∆)+(−∆)γ )(t−τ)S(τ)dτ (A.11)

θεn+1(t) = e(ε(−∆)+(−∆)γ )tθ0 +

∫ t

0e(ε(−∆)+(−∆)γ )(t−τ) (∇ · (v θεn)(τ) + S(τ)) dτ. (A.12)

Since the operatorexp ((ε(−∆) + (−∆)γ)t) is given explicitly by the Fourier multiplier withnon-zerosymbolexp ((ε|k|+ |k|γ)t), this operator does not alter the frequency support of the function it acts on.Therefore, it follows directly from Proposition 6.1, our assumptions on the frequency support onθ0 andS,thatsupp(θε1(t, ·)) ⊂ Pq for all t ∈ [0, T ). We proceed inductively and note that ifsupp(θεn) ⊂ Pq, then by

our assumption on the frequency support ofv and Proposition 6.1 we also havesupp(v θεn) ⊂ Pq. Hence,

we obtain, as in the casen = 0, thatsupp(θεn+1(t, ·)) ⊂ Pq for all t ∈ [0, T ), concluding the proof of theinduction step. This proves that the frequency support of all the iteratesθεn lies onPq.

It is left to prove that the sequenceθεnn≥1 converges to a functionθε which lies in the smoothness class(A.6). Note that there is no cancellation of the highest order term in the nonlinearity:

∫Λs(v ·∇θεn)Λsθεn+1.

However, since at least for nowε > 0 is fixed, we may use the full smoothing power of the Laplacian. First,note that from (A.4) it follows that for anyt ∈ (0, T ] we have

R1(t) := sup[0,t]

‖Λsθε1(τ, ·)‖2L2 +

∫ t

0‖Λs+γθ1(τ, ·)‖2L2dτ + ε

∫ t

0‖Λs+1θ1(τ, ·)‖2L2dτ

≤ ‖Λsθ0‖2L2 +

∫ t

0‖Λs−γS(τ, ·)‖2L2dτ. (A.13)

Hence, there exists a timeT1 ∈ (0, T ] such thatR1(T1) ≤ 2‖Λsθ0‖2L2 , e.g., anyT1 such that

T1 ≤‖Λsθ0‖2L2

‖S‖2L∞(0,T ;Hs−γ)

(A.14)

is sufficient. We proceed inductively, and assume that

Rn(t) := sup[0,t]

‖Λsθεn(τ, ·)‖2L2 +

∫ t

0‖Λs+γθn(τ, ·)‖2L2dτ + ε

∫ t

0‖Λs+1θn(τ, ·)‖2L2dτ (A.15)

is such thatRn(T1) ≤ 2‖Λsθ0‖2L2 . We now show that ifT1 is chosen appropriately, in terms ofε, θ0, v andS, we haveRn+1(T1) ≤ 2‖Λsθ0‖2L2 too. From (A.9), the fact that∇ · v = 0, integration by parts, ands > d/2 + 1− γ > d/2 which makesHs an algebra, we obtain

1

2

d

dt‖Λsθεn+1‖2L2 + ‖Λs+γθεn+1‖2L2 + ε‖Λs+1θεn+1‖2L2

≤ ‖Λs(vθεn)‖L2‖Λs+1θεn+1‖L2 + ‖Λs+γθεn+1‖L2‖Λs−γS‖L2

≤ 1

2ε‖Λsv‖2L2‖Λsθεn‖2L2 +

ε

2‖Λs+1θεn+1‖2L2 +

1

2‖Λs+γθεn+1‖2L2 +

1

2‖Λs−γS‖2L2 (A.16)

22 SUSAN FRIEDLANDER, WALTER RUSIN, AND VLAD VICOL

from which it follows that

Rn+1(T1) ≤ ‖Λsθ0‖2L2 +1

ε

∫ T1

0‖Λsv(τ, ·)‖2L2‖Λsθεn(τ, ·)‖2L2dτ +

∫ T1

0‖Λs−γS(τ, ·)‖2L2dτ

≤ ‖Λsθ0‖2L2 + T1

(2

ε‖v‖2L∞(0,T ;Hs)‖Λsθ0‖2L2 + ‖S‖2L∞(0,T ;Hs−γ)

). (A.17)

Hence, it follows from (A.13) that if we let

T1 =ε‖Λsθ0‖2L2

4‖v‖2L∞(0,T ;Hs)‖Λsθ0‖2L2 + ε‖S‖2L∞(0,T ;Hs−γ)

(A.18)

then we haveRn+1(T1) ≤ 2‖Λsθ0‖2L2 . Since the choice ofT1 cf. (A.18) is independent, ofn, it is clear thatthe inductive argument may be carried through, and henceRn(T1) ≤ 2‖Λsθ0‖2L2 independently ofn ≥ 1.To pass to a limit inn, we consider the difference of two iterates, and note that

∂t(θεn+1 − θεn) + (−∆)γ(θεn+1 − θεn)− ε∆(θεn+1 − θεn) + v · ∇(θεn − θεn−1) = 0 (A.19)

(θεn+1 − θεn)(0, ·) = 0 (A.20)

for all n ≥ 2. Similarly to (A.16), it follows from (A.19) that

Rn(t) := sup[0,t]

‖Λs(θεn+1 − θεn)(τ, ·)‖2L2 +

∫ t

0‖Λs+γ(θn+1 − θεn)(τ, ·)‖2L2dτ

+ ε

∫ t

0‖Λs+1(θn+1 − θεn)(τ, ·)‖2L2dτ

≤ 2

ε

∫ t

0‖Λsv(τ, ·)‖2L2‖Λsθ(τ, ·)‖2L2dτ (A.21)

and therefore

Rn(T1) ≤2T1ε

‖v‖2L∞(0,T ;Hs)Rn−1(T1) (A.22)

for all n ≥ 2. Thus, due to our choice ofT1 cf. (A.18), we have that

T1 ≤ε

4‖v‖2L∞(0,T ;Hs)

(A.23)

and henceRn(T1) ≤ R(T1)/2, which implies that the sequenceθεnn≥1 is not just bounded, but also acontraction in

L∞(0, T1;Hs) ∩ L2(0, T1;H

s+γ) ∩ εL2(0, T1;Hs+1). (A.24)

Hence there exists a limiting functionθε in the category (A.24). In addition, since for everyn ≥ 1 we havesupp(θεn) ⊂ Pq, and the setPq is closed (and even discrete), we automatically obtain thatsupp(θε) ⊂ Pq.

To show thatθε may be continued in the category (A.24) up to timeT , we note that‖Λsθε(T1)‖2L2 ≤2‖Λsθ0‖2L2 , and hence repeating the above argument with initial condition θε(T1), we obtain a solutionθε ∈ L∞(0, T1 + T2;H

s) ∩ L2(0, T1 + T2;Hs+γ) ∩ εL2(0, T1 + T2,H

s+1), where

T2 =2ε‖Λsθ0‖2L2

8‖v‖2L∞(0,T ;Hs)‖Λsθ0‖2L2 + ε‖S‖2L∞(0,T ;Hs−γ)

(A.25)

which is such that‖Λsθε(T1 + T2)‖2L2 ≤ 4‖θ0‖2L2 . Since the series

∑

k≥0

2kε‖Λsθ0‖2L2

2k+2‖v‖2L∞(0,T ;Hs)‖Λsθ0‖2L2 + ε‖S‖2L∞(0,T ;Hs−γ)

(A.26)

diverges for every fixedε > 0, the above argument may be terminated after finitely many steps, concludingthe construction of the solutionθε in the category (A.6).

SUPERCRITICALLY DIFFUSIVE(MG) 23

In order to conclude the proof of the lemma we need to pass to a limit as ε → 0. By construction wehave thatθε is uniformly bounded, with respect toε, inL∞(0, T ;Hs)∩L2(0, T ;Hs+γ), and from (A.4) weobtain that∂tθε is uniformly bounded, with respect toε, in L2(0, T ;Hs−2+γ). Thus, by the Aubin-Lionscompactness lemma (see for instance [25]), we obtain a weak limit θ ∈ L∞(0, T ;Hs) ∩ L2(0, T ;Hs+γ),so that the convergenceθε → θ is strong inL2(0, T ;Hs). Since the evolution is linear ands is largeenough, it follows that this limiting function is the uniquesolution of (A.1) which lies inL∞(0, T ;Hs) ∩L2(0, T ;Hs+γ). Lastly, since for everyε > 0 we havesupp(θε) ⊂ Pq, and sincePq is closed, we obtainthat the limiting function also has the desired support property, i.e. supp(θ) ⊂ Pq, which concludes theproof of the theorem.

Acknowledgements.The work of SF is supported in part by the NSF grant DMS-0803268. We are gratefulto Pierre Germain for stimulating discussions on an early version of this paper.

REFERENCES

[1] F. Bernicot and P. Germain. Bilinear oscillatory integrals and boundedness for new bilinear multipliers.Adv. Math.,225(4):1739–1785, 2010.

[2] L.A. Caffarelli and A. Vasseur. Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation.Ann. ofMath. (2), 171(3):1903–1930, 2010.

[3] D. Chae, P. Constantin, D. Cordoba, F. Gancedo, and J. Wu. Generalized surface quasi-geostrophic equations with singularvelocities.arXiv:1101.3537v1 [math.AP], 2011.

[4] D. Chae, P. Constantin, and J. Wu. Inviscid models generalizing the 2D Euler and the surface quasi-geostrophic equations.Arch. Ration. Mech. Anal., 202(1):35–62, 2011.

[5] P. Constantin and J. Wu. Behavior of solutions of 2D quasi-geostrophic equations.SIAM J. Math. Anal., 30(5):937–948, 1999.[6] A. Cordoba and D. Cordoba. A maximum principle appliedto quasi-geostrophic equations.Comm. Math. Phys., 249(3):511–

528, 2004.[7] D. Ebin. Ill-posedness of the Rayleigh-Taylor and Helmholtz problems for incompressible fluids.Comm. Partial Differential

Equations, 13(10):1265–1295, 1988.[8] S. Friedlander, W. Strauss, and M. Vishik. Nonlinear instability in an ideal fluid.Ann. Inst. H. Poincare Anal. Non Lineaire,

14(2):187–209, 1997.[9] S. Friedlander and V. Vicol. Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dy-

namics.Ann. Inst. H. Poincare Anal. Non Lineaire, 28(2):283–301, 2011.[10] S. Friedlander and V. Vicol. On the ill/well-posednessand nonlinear instability of the magneto-geostrophic equations.Non-

linearity, 24(11):3019–3042, 2011.[11] D. Gerard-Varet and E. Dormy. On the ill-posedness of the Prandtl equation.J. Amer. Math. Soc., 23(2):591–609, 2010.[12] Y. Guo and T. Nguyen. A note on the Prandtl boundary layers.arXiv:1011.0130v3 [math.AP], 2010.[13] T. Hillen and K.J. Painter. A user’s guide to pde models of chemotaxis.J. Math. Biol, 58:183–217, 2009.[14] N. Ju. Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space.Comm.

Math. Phys., 251(2):365–376, 2004.[15] C.E. Kenig, G. Ponce, and L. Vega. Well-posedness of theinitial value problem for the Korteweg-de Vries equation.J. Amer.

Math. Soc., 4(2):323–347, 1991.[16] A. Kiselev, F. Nazarov, and A. Volberg. Global well-posedness for the critical 2D dissipative quasi-geostrophic equation.

Invent. Math., 167(3):445–453, 2007.[17] W. McLean. Local and global descriptions of periodic pseudodifferential operators.Math. Nachr., 150:15-161, 1991.[18] L.D. Mesalkin and J.G. Sinaı. Investigation of the stability of a stationary solution of a system of equations forthe plane

movement of an incompressible viscous liquid.J. Appl. Math. Mech., 25:1700–1705, 1961.[19] H.K. Moffatt. Magnetostrophic turbulence and the geodynamo. InIUTAM Symposium on Computational Physics and New

Perspectives in Turbulence, volume 4 ofIUTAM Bookser., pages 339–346. Springer, Dordrecht, 2008.[20] M. Renardy. Ill-posedness of the hydrostatic Euler andNavier-Stokes equations.Arch. Ration. Mech. Anal., 194(3):877–886,

2009.[21] W. Rusin. Logarithmic spikes of gradients and uniqueness of weak solutions to a class of active scalar equations,

arXiv:1106.2778v1 [math.AP], 2011.[22] R. Shvydkoy. Convex integration for a class of active scalar equations.J. Amer. Math. Soc., 24(4):1159–1174, 2011.[23] E.M. Stein.Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 ofPrinceton

Mathematical Series. Princeton University Press, Princeton, NJ, 1993.[24] T. Tao.Nonlinear dispersive equations, volume 106 ofCBMS Regional Conference Series in Mathematics. Published for the

Conference Board of the Mathematical Sciences, Washington, DC, 2006. Local and global analysis.

24 SUSAN FRIEDLANDER, WALTER RUSIN, AND VLAD VICOL

[25] R. Temam.Navier-Stokes equations: Theory and numerical analysis. AMS Chelsea Publishing, Providence, RI, 2001. Reprintof the 1984 edition.

[26] J. Wu. Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces.SIAM J. Math. Anal., 36(3):1014–1030 (electronic), 2004/05.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF SOUTHERN CALIFORNIA , 3620 S. VERMONT AVE., LOS ANGELES,CA 90089

E-mail address: susanfri@usc.edu

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF SOUTHERN CALIFORNIA , 3620 S. VERMONT AVE., LOS ANGELES,CA 90089

E-mail address: wrusin@usc.edu

DEPARTMENT OFMATHEMATICS, THE UNIVERSITY OF CHICAGO, 5734 UNIVERSITY AVE., CHICAGO, IL 60637E-mail address: vicol@math.uchicago.edu